Homogenisation: Averaging Processes in Periodic Media

Sections 1 and 2 deal with the classical and generalized formulations of boundary value problems for the heat equation; §3 presents the generalized formulations of problems for the basic equations of mathematical physics and states the existence and uniqueness theorems for generalized solutions. These questions are dealt with in more detail in the monographs [74–76, 78–80, 147, 151, 155] and articles [43, 51, 77, 104, 105, 171].

This Chapter introduces definitions of an asymptotic expansion of a function, operations with asymptotic series, and discusses elementary examples of the methods of asymptotic expansions applied to the problem of averaging the processes in periodic media. Many model examples of asymptotic methods used for investigating problems of mechanics are discussed in [101, 103].

The averaging method is applied in this chapter to investigations of some spatial one-dimensional problems. The chapter considers the elastic rod, the heat equation, nonlinear equations, the process of heat transfer in a system of screens separated by interlayers, and equations with random coefficients.

In this chapter we will consider partial differential equations which describe physical processes in composite materials. Section 1 of the Introduction cited an example of thermal conductivity problems (l)–(3) for a fibrous composite (Fig. 1 shows its section by a plane perpendicular to the fibers, ε is the side of a periodic cell, ε ≪ 1). By analogy with §3 of the Introduction let us introduce the “fast” variables ξ _q = x _q / ε. Then the elementary cell—the square (0, ε) x (0, ε) in the variables x ₁, x ₂—turns into the unit square in the variables ξ₁, ξ₂ (see Fig. 2). Again we denote by <Emphasis FontCategory=“NonProportional”>B</Emphasis> the domain in the unit square which corresponds to the fiber cross-section and by <Emphasis FontCategory=“NonProportional”>M</Emphasis> the domain occupied by the matrix. Let where K (ξ₁, ξ₃) is a periodic function with period 1 in ξ₁ and ξ₂. Then the conductivity coefficient K _ε (x₁, x₂) in §1 of the Introduction can be represented as K _ε (x₁, x₂) = K (x_1/ε, x₂/ε). Note that K (x₁, x₂) s a periodic function of x₁ and x₂ with period ε.

This chapter investigates properties of solutions of equations on a cell and of averaged equations and relates the average characteristics of the original problem to those of a solution of the averaged equation.

This chapter presents mathematical models of composite materials where elastic, thermal, or other physical characteristics of the reinforcement differ greatly from those of the matrix. High strength and rigidity and good thermal insulation properties of these materials have made them suitable for many applications in engineering today (see [52, 66, 91, 146, 149, 150]). The mathematical modelling of composites with high-modulus reinforcement is distinguished above all by the presence of the large dimensionless parameter ω, which is the order of the ratio of the reinforcement’s particular characteristic to that of the matrix. Thus, for plastics reinforced with fibers of carbon, boron, or silicon carbide, the ratio of Young’s modulus of the reinforcement E _B to that of the matrix E _M is of order 100 (see [52]). There is a great distinction between the characteristics of air and the matrix in porous media, such as rubber and metals. This chapter is concerned mainly with obtaining explicit formulas for the effective characteristics of these materials in the multi-dimensional case (see the principle of splitting in §5).

Frameworks have been widely adopted in engineering today, primarily for building structures (trusses of bridges and industrial installations, frameworks of housing, supports of electric power lines, etc.). The methods of solving the system of equations of the strength of materials theory presently used for calculating the frameworks have the drawback that the number of equations increases with the number of nodes of the skeletal structure. This greatly impedes the calculation of skeletal structures with a large number of nodes, especially when investigating nonstationary and nonlinear processes.

All foregoing sections, except §§7.3 and 8.6, ignored the boundary effects that may occur near the boundary of a composite (the investigations covered only the zeroth approximation of the boundary value problem without oscillation under boundary condition). Yet the neighbourhoods of the boundaries and contact surfaces (when two pieces of different composites are in contact) have boundary layers where the gradient of a solution may greatly differ (by a value of order 1) from those of the asymptotics obtained above.